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Questions tagged [numerical-methods]

Questions on numerical methods; methods for approximately solving various problems that often do not admit exact solutions. Such problems can be in various fields. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.

In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems.

Definitions: Numerical methods are techniques to approximate mathematical procedures (example of a mathematical procedure is an integral).

Approximations are needed because we either cannot solve the procedure analytically (example is the standard normal cumulative distribution function) or because the analytical method is intractable (example is solving a set of a thousand simultaneous linear equations for a thousand unknowns for finding forces in a truss).

Applications: With the advent of the modern high speed electronic digital computers, the numerical methods are successfully applied to study problems in mathematics, engineering, computer science and physical sciences such as biophysics, physics, atmospheric sciences and geo-sciences.

Possible topics include but are not limited to:

  1. Approximation theory, interpolations.
  2. Numerical ODE/PDE.
  3. Root finding algorithm.
  4. Numerical linear algebra, matrix computations.
  5. Discrete integral transform, FFT, etc.
  6. Linear/Non-linear programming, integer optimization.

For questions concerning matrices, please consider adding the tag.

For questions concerning optimization, please consider adding the tag.

For questions concerning Numerical ODE/PDE, please consider adding the // tag.

References:

https://en.wikipedia.org/wiki/Numerical_method

"Numerical Methods for Scientific and Engineering Computation" by M. K. Jain, S.R.K. Iyengar, R. K. Jain

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Euler's method, no interval

I have a problem on which I need to apply Euler's method - EXCEPT that I don't have one of the crucial components. Question and my thoughts below: Question: Consider the initial value problem $\frac{dy}{dt}=\alpha t^{\alpha - 1}, y(0)=0$, where…
user85362
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Gauss-Newton Method

Can anyone give some guidance with the following question:- Prove that the Gauss-Newton Method applied to a linear system $Ax=b$ converges in one step to the solution of the normal equations
user76020
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Newton's method for a given polynomial

Let $f(x)=\frac{1}{5}x^5+\frac{1}{3}x^3+x-1$ Show that $f$ has only one zero $r$ in interval $(0,1)$ To find approximation of $r$ we apply Newton's method $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$. Show that for every $x_0\in(0,1)$ this method…
xan
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4th order Runge-Kutta for Coupled ODEs

here is my code: tmax = 100; % Some Arbitrary Value tmin = 0; % Start at t = 0 n = (tmax - tmin)/h; t = linspace(tmin, tmax, n); Cs = zeros(n,1); Ce = zeros(n,1); Ces = zeros(n,1); Cp = zeros(n,1); %% % k1, k2, k3 are constants in the…
user105210
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Bisection Method for intersection of two functions

I know how to use the bisection method when finding roots, however I don't know how to use it for when two lines intersect, any help with this would be much appreciated
Justin
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Find a way to calculate $f(x) = x-\sqrt{x^2-\alpha}, \alpha << x.$

I thought about this but I could not come up with a way to calculate this. Any comment?? note:$x$ is much larger than $\alpha$
eChung00
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Alternative form for sinh(x)/cosh(x) .

I have the following expression. $$ \tanh (x) = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}} $$ I know that they derive one from another , but how do I rewrite them in alternative forms in order to not get a NaN(not a number) when…
rhauder_
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I need to solve this equation numerically but don't have access to any software...

I need to solve this equation numerically, but I don't have access to any software: $$2^{1-t} \log \tfrac12 = a(t) \log a(t)$$ where $a(t):= 2^{-t} + \tfrac12$. The logs are natural logarithms. Thanks for your help!
Frank
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Numerical Analysis Using Nonlinear Least-Square Fit function of the form f(t) = Asin(Bt) + Ccos(Dt) to fit 24 data points

can anybody help me solve this problem. As I am stuck at the $E(A,B,C,D)=\sum_{k=1}^{24} (f(t_k)-y_k)^2 $ also $E_2 (f)=? $ is the root mean square error I am having difficulty finding the correct bestfitcurve as I arbitraririly chose initial…
user182936
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Newton's Method - iteration formula

The iteration formula $$x_{n+1}=x_n-(\cos x_n)(\sin x_n)+R\cos^2x_n$$ where $R$ is a positive constant is obtained by applying Newton's method to some function $f(x)$. What is $f(x)$? What can this formula be used for? I got the solution for $f(x)$…
Drake
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Application of Newton-Raphson method

Is Newton-Raphson method applicable to solve rational equations? or any equation except polynomial equations.
Sasan
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Precision of $\sinh x$ at $x \approx 0$

What can we do to increase the precision of $\sinh x$ at $x \approx 0$? I tried to use conjugation , but it gives me $e^{2x} - e^{-2x}$ which is again the subtraction of "similar" numbers at $x \approx 0$ Thank you
John Lennon
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What is a function that is discontinuous, yet the bisection method converges?

Also what about a function for which it diverges? I was thinking f(x) = 10 + 1/x would work for where it converges, but can't think of one where the bisection method diverges at near a root. Thanks for any thoughts!
Drake
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Problem related to fixed point iteration

How can I use fixed point iteration to solve $x^2 = 3$ using $g(x) = x^2 + x - 3$ to find the numerical value of the solution $x = +\sqrt{3}$. What happens? Then I use $g(x) = (x + 3/x)/2$. For which values of $x_0$ is this guaranteed to converge to…
user87274
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